Here's something that may help you a lot, once I realized this, things made a lot better sense to me:
Current in a circuit is (generally) not "set" by the power supply, it is set by the load (resistor, motor, hair dryer, etc)
The load "pulls" or "draws" current. Think of household wiring. You may have an outlet in your bathroom on a 15 amp breaker. Let's say a hair dryer on high "pulls" 10 amps. Now if you plug in TWO hair dryers on high, they would attempt to draw 20 amps, thus tripping the 15 amp breaker.
Now, power supplies have a maximum amount of current they can deliver, as you saw in your huge 9v/LED experiment. The LED's wanted more, but the 9v simply couldn't deliver. BTW, you probably need a new 9v now. Better yet, toss that thing and use better batteries (like some AA, C, or D in series, or a wall wort). Batteries have an "Amp Hour" or "Milliamp Hour" (mAh) rating that you can use to actually calculate how long a battery will power your circuit!
Note that LED's are not the same as a light bulb though. Without a resistor, an LED will pull so much current that it will burn itself up.
When learning electronics, if you don't blow some stuff up here and there you're not learning properly LOL